Page 1. Before-After Control-Impact (BACI) Power Analysis For Several Related Populations (Variance Known) Richard A. Hinrichsen. September 24, 2010

Transcription

1 Pag for-aftr Cotrol-Impact (ACI) Powr Aalysis For Svral Rlatd Populatios (Variac Kow) Richard A. Hirichs Sptmbr 4, Cavat: This primtal dsig tool is a idalizd powr aalysis built upo svral simplifyig assumptios (Tabl ). For a spcific primt, a mor accurat portrayal of powr may rquir chagig ths assumptios ad th udrlyig quatios. (For ampl, assumig that variac will b stimatd istad of kow.) Thrfor this aalysis should b tratd as a rough guid to powr. Itroductio Currtly thr ar may watrshd projcts udrway i th Columbia asi to dtrmi th survival ffcts of various maagmt actios o salmo survival. For ampl, thr ar a sris of itsivly moitord watrshds (IMWs) big stablishd for th purpos of bttr udrstadig how salmo rspod to approachs to rstor habitat. Wh ths projcts ar ru as primts, it is possibl to idtify th ffctivss of rstoratio ad othr maagmt actios. This aalysis was motivatd by th d to dsig ths primts i such a way that thy hav a good chac to dtct sigificat survival chags i salmo wh thy occur. Usig this tool ca giv a primtr a rough ida of th umbr of yars to ru a primt ad dtrmi what statistical powr ca b pctd basd o what is kow about th variaccovariac structurs for survival amog th salmo populatios studid. Th framwork for th aalysis giv hr, although dvlopd with salmo i mid, fits ito th framwork of th for-aftr-cotrol-impact (ACI) primt. Such ACI-typ primts fid applicatio byod Columbia Rivr salmo survival (Osbrg ad Schmitt 996).

2 Pag A a priori powr aalysis is dvlopd for a ACI-typ primt aimd at stimatig a chag i survival for svral populatios. Th primt icluds a for priod whr all populatios rciv o tratmt followd by a Aftr priod whr oly th tratmt populatios rciv tratmt. This is a gralizatio of th ACI-typ primt whr th cotrol populatio ad impact populatio ar sampld o tim bfor ad o tim aftr th tratmt (Gr 979, Osbrg ad Schmitt 996). Th assumptios for th aalysis ar giv i Tabl. It is assumd that i th absc of tratmt, all populatios hav a commo ma log survival. caus this assumptio ad othrs may ot hold i practic, this aalysis should b tratd as a rough guid. Th mai goal of this work is to dmostrat th probability of dtctig a ffct o survival wh svral rlatd populatios with a commo ma survival ar usd i a ACI-typ primt. This goal is accomplishd by dscribig th primt i a statistically rigorous way, sttig up th liklihood fuctio, ad th usig maimum liklihood thory to stimat powr. Powr is th probability of rjctig th ull hypothsis of o tratmt ffct. A A A3 A4 A5 A6 A7 Tabl. Assumptios usd i powr aalysis. Th obsrvatios of log(survival) follow a multivariat ormal distributio. Thr is o srial dpdc. All populatios hav a commo ma log(survival) bfor tratmt. Aftr tratmt, th cotrol populatios cotiu to hav th sam commo ma as hibitd i th for yars, ad th tratmt populatios also hav a commo ma, but shiftd by a amout (th ffct siz) that is th sam for all tratmt populatios. Th masurmt rrors i log(survival) follow a multivariat ormal distributio ad th rrors ar idpdt of th rror du to actual yar-to-yar viromtal variability. Th stimator of th tratmt ffct is a maimum liklihood stimat. Th variac-covariac matri rprstig th rror i log(survival) is kow. Ths ar assumptios for a idalizd primt. For a spcific applicatio, a mor accurat primtal dsig may rquir chagig ths assumptios ad th udrlyig quatios. Thrfor this aalysis should b tratd as a rough guid to powr. Th wbsit cotais a wb-basd tool that implmts this powr aalysis with th addd assumptios that th variacs i log(survival) ar qual for all populatios ad th corrlatios i log(survival) ar qual for ach pair of populatios. This is th itraclass covariac structur studid by R.A. Fishr (95). Th R-cod for implmtig this powr aalysis may b foud i Appdi A (Vrabls t al. ).

3 Pag 3 Mthods To coduct th powr aalysis, a modl was formulatd ad maimum liklihood stimators wr drivd (Mood t al. 974). Ths stimators wr th usd as th basis for tstig th ull hypothsis of o tratmt ffct usig Mot Carlo simulatio. Powr was th calculatd as th probability that th ull hypothsis is rjctd. Th modl. It was assumd that ma log(survival) bfor tratmt was th sam for ach populatio ad qual to. Aftr tratmt, th ma log(survival) of th tratmt populatios shifts by th amout for th tratmt populatios whil th cotrol populatios cotiu to hav a ma log(survival) of. It was also assumd that yar-to-yar variability i log(survival) ad masurmt rror followd a multivariat ormal distributio with variac y m, whr y is th variaccovariac matri that dscribs yar-to-yar variability i th absc of masurmt rror, ad m rprsts th variac-covariac matri of th masurmt rror. I this implmtatio of th ACI primt, th variac-covariac matri,, was assumd to b kow. Maimum liklihood stimators. To driv maimum liklihood stimators, a prssio for th liklihood fuctio is dd. For th modl dscribd abov, th logliklihood fuctio is l ( θ, ) C ( / )l (/ ) t ( t θ) ( θ)) t () θ) ( ) ( / ) ( t θ t t ; whr θ ; is th variac-covariac matri; C is a costat that dos ot dpd o th paramtrs; is th umbr of yars prior to tratmt; is th total umbr of yars of th primt; t is a k-vctor of obsrvd survivals i yar t; k is th umbr of populatios (tratmt + cotrol) usd i th primt; is a k-vctor of s; is a k -vctor of k s followd by k s, whr k rprsts th umbr of cotrol populatios ad krprsts th umbr of tratmt populatios. Th vctor t is arragd so that th k cotrol populatios prcd th k tratmt populatios.

4 Pag 4 Usig maimum liklihood thory, stimatig quatios for th for ma, tratmt ffct ar dvlopd. I this problm, maimizig th liklihood fuctio is quivalt to a solvig th gralizd last squars problm of miimizig SS ; () whr rprsts th k-vctor of sampl mas of log(survival) i th for priod, ad rprsts th k-vctor of sampl mas of log(survival) i th Aftr priod. This gralizd sum of squars may b writt i th familiar form θ y θ y SS ; (3) whr y,, ad. I this form, th gralizd last squars solutio, th calld th Aitk stimator (Prss 5), is kow to b y θ T T. (4) Aftr cosidrabl matri algbra, w may writ

5 Pag 5 ) ( ) ( θ μ, (5) whr is a k-vctor rprstig populatio-spcific sampl mas ovr th tir duratio of th primt. Also wll kow is th coditioal variac of th stimat of θ (giv that th variac-covariac matri): var θ T. (6) I particular, th variac of th tratmt ffct stimator is var. (7) This formula shows how th variac of th tratmt ffct stimat dpds o th valus of k, k,, ad th tris of th variac-covariac matri. Usig this formula, it is th simpl to calculat th stadard rror of th tratmt ffct stimat, which is qual to th squar root of th variac: ). var( ) ( s (8)

6 Pag 6 Th cofficit of variatio is CV ( ) s( ) /. (9) Powr. Powr is th probability of rjctig th ull hypothsis of o tratmt ffct wh th actual tratmt ffct is. Powr dpds o th tru tratmt ffct, th probability of a typ I rror (usually calld th alpha valu), ad th stadard rror of th stimator. y maimum liklihood thory th stimator of th tratmt ffct is asymptotically ormally distributd. I this cas it may b show that th stimator is ormally distributd rgardlss of th sampl siz. This occurs bcaus th variac-covariac matri is assumd kow ad th stimator is a liar combiatio of radom variabls that ar kow to follow a multivariat ormal distributio. Ay liar combiatio of ormally distributd radom variabls is also ormally distributd. Thus, th powr may b writt as ) ( z / s( )) ( z / s( )); () ( / / whr (z) is th cumulativ distributio fuctio of a radom variabl that follows a stadard ormal distributio (a ormal distributio with ma zro ad stadard dviatio ), ad z / is th critical valu such that / probability lis to th right of th valu z / i a stadard ormal distributio. For ampl, wh. 5, th critical valu is qual to.96. Eprimtrs oft choos a dsig such that powr of.8 is achivd.

7 Pag 7 Ackowldgmt This work was supportd by ovill Powr Admiistratio cotract # Thaks to Charli Pauls, Rishi Sharma, ad Tracy Hillma for thir valuabl rviws. Thaks to ria Maschhoff for implmtig this aalysis as a wb tool at

8 Pag 8 Rfrcs Fishr, R.A. 95. Statistical Mthods for Rsarch Workrs. Olivr ad oyd, Ediburgh, Scotlad. Gr, R.H Samplig dsig ad statistical mthods for viromtal biologists. Wily ad Sos, Nw York, Nw York. Mood, A.M, Graybill, F.A., ad D.C. os Itroductio to th thory of statistics, Third Editio. McGraw-Hill, Nw York, Nw York. Osbrg, C.W. ad R.J. Schmidt Dtctig cological impacts causd by huma activitis. I Dtctig Ecological Impacts: Cocpts ad Applicatios i Coastal Habitats, R.J Schmitt ad C.W. Osbrg, Editors. Acadmic Prss, Nw York, Nw York. Prss, S.J. 5. Applid multivariat aalysis: usig aysia ad frqutist mthods of ifrc. Dovr Publicatios, Ic. Miola, Nw York Vrabls, W.N., Smith, D.M., ad R Dvlopmt Cor Tam.. A Itroductio to R. Nots o R: A Programmig Eviromt for Data Aalysis ad Graphics Vrsio.. (-5-3).

9 Pag 9 Appdi A. R cod usd to calculat statistical powr for th ACItyp primt wh variac is kow #Program to stimat stadard rrors ad powr i a ACI-typ primt #s is yar-to-yar variac (assumd qual for all populatios) #rho is th corrlatio of survivals btw ach pair of populatios # umbr of bfor yars # umbr of aftr yars #k umbr of cotrol populatios #k umbr of tratmt populatios #m masurmt rror #alpha -- prob. typ I rror (Probability of rjctig ull hypothsis wh tru.) #dlta -- tru tratmt ffct rprstig diffrc i atural log survival #l(stratmt/scotrol) baci<-fuctio(s=,rho=.9,=5,=5,k=,k=,m=log(.),alpha=.5,dlta=log(.5)){ k<-k+k <-+ SIG<-matri(s*rho,col=k,row=k) diag(sig)<-s+m*m INVSIG<-solv(SIG) <-rp(,k) s<-*t()%*%invsig%*% <-c(rp(,k),rp(,k)) dt<-*t()%*%invsig%*% dt<-dt**t()%*%invsig%*%-**(t()%*%invsig%*%)^ s<-sqrt(s/dt) #rul--rjct wh stimat cds.96 ss i absolut valu (two-sidd) q<-qorm(-alpha/) powr<-(-porm(q*s,ma=dlta,sd=s))+porm(-q*s,ma=dlta,sd=s) rtur(list(s=s,rho=rho,=,=,k=k,k=k,m=m, alpha=alpha,dlta=dlta, s=s,cv=s/dlta,powr=powr)) } #outputs #s -- stadard rror #cv -- cofficit of variatio #powr -- probability of rjctig th ull hypothsis of o ffct #This R-cod uss th addd assumptios of a commo variac ad commo corrlatio #trms (th itraclass covariac structur studid by R. A. Fishr (95). This assumptio may #b rlad by spcifyig SIG as a iput to th fuctio baci i plac of th iput variabls s #ad rho.